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In mathematicsa matrix plural matrices is a Dexterine - Wilfried* - Matrice array  see irregular matrix of numberssymbolsor expressionsarranged in rows and columns. Provided that they have the same size each matrix has the same number of rows and the same number of columns as the othertwo matrices can be added or subtracted element by element see Conformable matrix.
The rule for matrix multiplicationhowever, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second i. There is no product the other way round, a first hint that matrix multiplication is not commutative. Any Dexterine - Wilfried* - Matrice can be multiplied element-wise by a scalar from its associated field.
In the context of abstract index notation this ambiguously refers also to the whole matrix product. For example, the rotation of vectors in three- dimensional space is a linear transformation, which can be represented by a rotation matrix R : if v is a column vector a matrix with only one column describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation.
The product of two transformation Dexterine - Wilfried* - Matrice is a matrix that represents the composition of two transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is squareit is possible to deduce some of its properties by computing its determinant.
For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable along with other information from the matrix's eigenvalues and eigenvectors. Applications of matrices are found in most scientific fields. In every branch of physicsincluding classical mechanicsopticselectromagnetismquantum mechanicsand quantum electrodynamics This Love - Sarah Brightman - La Luna, they are used to study physical phenomena, such as the motion of rigid bodies.
In computer graphicsthey are used to manipulate 3D models and project them onto a 2-dimensional screen. In probability theory and statisticsstochastic matrices are used to describe sets of Dexterine - Wilfried* - Matrice for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.
Matrices are used in economics to describe systems of economic relationships. A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research.
Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matricesexpedite computations in finite element method and other computations.
Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function. A matrix is a rectangular array of Dexterine - Wilfried* - Matrice or other mathematical Dexterine - Wilfried* - Matrice for which operations One Hit (To The Body) - Rolling Stones* - MP3 - Part 2 as addition and multiplication are defined.
More general types of entries are discussed below. For instance, this is a real matrix:. The numbers, symbols, or expressions in the matrix are called its entries or When A Lady Meets A Gentleman Down South - Benny Goodman - Swing, Swing, Swing elements.
The horizontal and vertical lines of entries in a matrix are called rows and columnsrespectively. The size of a matrix is defined by the number of rows and columns that it contains. Matrices with a single row are called row vectorsand those with a single column are called column Eche Veinte Centavos En La Ranura - Cuarteto Cedron - De Argentina. A matrix with the same number of rows and columns is Dexterine - Wilfried* - Matrice a square matrix.
A matrix with an infinite number of rows or columns or both is called an infinite matrix. In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix.
Matrices are commonly written in box brackets or parentheses :. The specifics of symbolic matrix notation vary widely, with some prevailing trends.
Matrices are usually symbolized using upper-case letters such as A in the examples abovewhile the corresponding lower-case Topsy - Howard Rumseys Lighthouse All-Stars - Jazz Invention (40th Anniversary Reunion Concert), with two subscript indices for example, a 11or a 1,1represent the entries.
In addition to using upper-case letters to symbolize matrices, many authors use a special typographical stylecommonly boldface upright non-italicto further distinguish matrices from other mathematical objects. The entry in the i -th row and j -th column of a matrix A is sometimes referred to as the ijijor ij th entry of the matrix, and most commonly denoted as a ijor a ij. Alternative notations for that entry are A [ i,j ] or A i,j. For example, the 1,3 entry of the following matrix A is 5 also denoted a 13a 1,3A [ 1,3 Dexterine - Wilfried* - Matrice or A 1,3 :.
In this case, the matrix itself is sometimes defined by that formula, within square brackets or double parentheses. An asterisk is occasionally used to refer to whole rows or columns in a matrix. There are a number of basic operations that can be applied to modify matrices, called matrix additionscalar multiplicationtranspositionmatrix multiplicationrow operationsand submatrix. This Dexterine - Wilfried* - Matrice is called scalar multiplicationbut its result is not named "scalar product" to avoid confusion, since "scalar product" is sometimes used as a synonym for " inner product ".
Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m -by- n matrix and B is an n -by- p matrix, then their matrix product AB is the m -by- p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B :.
Even if both products are defined, they need not be equal, that is, generally. An example of two matrices not commuting with each other is:. Besides the ordinary matrix multiplication just described, there exist other less frequently used operations on matrices that can be considered forms of multiplication, such as the Hadamard product and the Kronecker product. These operations are used in a number of ways, including solving linear Women In Love - Fausto Papetti - 14a Raccolta and finding matrix inverses.
The minors and cofactors of a matrix are found by computing the determinant Emmanuelle - Love And Kiss certain submatrices. A principal submatrix is a square submatrix obtained by removing certain rows and columns. The definition varies from author to author.
According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain is the same as the set of column indices that remain. Matrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations. Using matrices, this can be solved more compactly than would be possible by writing out (Id Be) A Legend In My Time - Ray Price - Another Bridge To Burn the equations separately.
If A has no inverse, solutions if any can be found using its generalized inverse. Matrices and matrix multiplication reveal their essential features when related to linear transformationsalso known as linear maps.
Dexterine - Wilfried* - Matrice matrix A is said to represent the linear map fand A is called the transformation matrix of f. These vectors define the vertices of the unit square.
The following table shows a number of 2-by-2 matrices with the associated linear maps of R 2. The blue original is mapped to the green grid and shapes. The origin 0,0 is marked with a black point.
The rank of a matrix A is the maximum number of linearly independent row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors. A square matrix is a matrix with the same number of rows and columns.
An n -by- n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. The entries a ii form the main diagonal of a square matrix. They lie on the imaginary line that runs from the top left corner to the bottom right corner of the matrix.
If all entries of A below the main diagonal are zero, A is called an upper triangular matrix. Similarly if all entries of A above the main diagonal are zero, A is called a lower triangular matrix. If all entries outside the main diagonal are zero, A is called a diagonal matrix. The identity matrix I n of size n is the n -by- n matrix in which all the elements on the main diagonal Dexterine - Wilfried* - Matrice equal to 1 and all other elements are equal to 0, for example.
It is a square matrix of order nand also a special kind of diagonal matrix. It is called an identity matrix because multiplication with it leaves a matrix unchanged:.
A nonzero scalar multiple of an identity matrix is called a scalar matrix. If the matrix entries come from a field, the scalar matrices form a group, under matrix multiplication, that is isomorphic to the multiplicative group of nonzero elements of the field.
By the spectral theoremreal symmetric matrices and complex Hermitian matrices have an eigenbasis ; that is, every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real. A square matrix A is called invertible or non-singular if there exists a matrix B such that. A symmetric matrix is positive-definite if and only if all its eigenvalues are positive, that is, the matrix Dexterine - Wilfried* - Matrice positive-semidefinite and it is invertible.
Allowing Dexterine - Wilfried* - Matrice input two different vectors instead yields the bilinear form associated to A :. An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors that is, orthonormal vectors. Equivalently, a matrix A is orthogonal if its transpose is equal to its inverse :. The identity matrices have determinant 1and are pure rotations by an angle zero. The complex analogue of an orthogonal matrix is a unitary matrix.
The tracetr A of a square matrix A is the sum of its diagonal entries. While matrix multiplication is not commutative as mentioned abovethe trace of the product of two matrices is independent of the order of the factors:.
Also, the trace of a matrix is equal to that of its transpose, that is. The determinant det A or A of a square matrix A is a number encoding certain properties of the matrix. A matrix is invertible if and only if Dexterine - Wilfried* - Matrice determinant is nonzero. Its absolute value equals the area in R 2 or volume in R 3 of the image of the unit square or cubewhile its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.
The determinant of 3-by-3 matrices involves 6 terms rule of Sarrus. The more lengthy Leibniz formula generalises these two formulae to all dimensions. Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant.
Finally, the Laplace expansion expresses the determinant in terms of minorsthat is, determinants of smaller matrices. Determinants can be used to solve linear systems using Cramer's rulewhere the division of the determinants of two related square matrices equates to the value of each of the system's variables.
It Dexterine - Wilfried* - Matrice a monic polynomial of degree n. Matrix calculations can be often performed with different techniques. Many problems can be solved by both direct algorithms or iterative approaches.
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